Dimension of open subsets of $R^n$

Does an open subset of $R^n$ exist that has dimension less than $n$ in the standard topology? All the less than $n$ dimensional subsets I can think of are not open.

Solution 1:

In $\mathbb{R}^n$, open balls are homeomorphic to the whole space (use $f(x) = \frac{1}{1+||x||}\cdot x$, for $\mathbb{R}^n$ and the open unit ball, and use translations and scaling to make all open balls homeomorphic among themselves.) This implies $\dim(B) = \dim(\mathbb{R}^n) = n$ (the latter is a deep theorem)

and dimension functions are monotone in metric spaces, so if a set $A$ contains some open ball $B$, then $\dim(B) \le \dim(A)$. So all open sets have dimension $n$.